complement pairs in a lattice
0
I think $a,b,c$ have no complements.
Please confirm. I am confused with the definition of the complement of elements of a lattice.
graph-theory lattice-orders
edited Nov 28 '18 at 17:03
Alex Vong
1,286819
asked Nov 28 '18 at 16:52
amitPamitP
11
add a comment |
0
I think $a,b,c$ have no complements.
Please confirm. I am confused with the definition of the complement of elements of a lattice.
graph-theory lattice-orders
edited Nov 28 '18 at 17:03
Alex Vong
1,286819
asked Nov 28 '18 at 16:52
amitPamitP
11
I agree with you that $a,b,c$ have no complements. A complement of an element $x$ in a lattice is an element $y$ such that $xlor y=hat{0}$ and $xland y=hat{1}$. In this lattice, $e$ is $hat{1}$ and $d$ is $hat{0}$. If $x$ is a complement of $a$, then we know that $x$ can't be $f$ or $e$, otherwise $xland a=a$. But we can't have $x$ be $b,c,$ or $d$ either, otherwise $xlor a=f$ if $x$ is $b$ or $c$ and $dlor a=a$. Thus, $a$ has no complements. The same argument applies to $b$ and $c$ by symmetry.
– Kevin Long
Nov 28 '18 at 18:33
add a comment |
0
0
0
I think $a,b,c$ have no complements.
Please confirm. I am confused with the definition of the complement of elements of a lattice.
graph-theory lattice-orders
edited Nov 28 '18 at 17:03
Alex Vong
1,286819
asked Nov 28 '18 at 16:52
amitPamitP
11
I think $a,b,c$ have no complements.
Please confirm. I am confused with the definition of the complement of elements of a lattice.
graph-theory lattice-orders
graph-theory lattice-orders
edited Nov 28 '18 at 17:03
Alex Vong
1,286819
asked Nov 28 '18 at 16:52
amitPamitP
11
edited Nov 28 '18 at 17:03
Alex Vong
1,286819
asked Nov 28 '18 at 16:52
amitPamitP
11
edited Nov 28 '18 at 17:03
Alex Vong
1,286819
edited Nov 28 '18 at 17:03
Alex Vong
1,286819
edited Nov 28 '18 at 17:03
Alex Vong
1,286819
1,286819
asked Nov 28 '18 at 16:52
amitPamitP
11
asked Nov 28 '18 at 16:52
amitPamitP
11
asked Nov 28 '18 at 16:52
amitPamitP
11
11
I agree with you that $a,b,c$ have no complements. A complement of an element $x$ in a lattice is an element $y$ such that $xlor y=hat{0}$ and $xland y=hat{1}$. In this lattice, $e$ is $hat{1}$ and $d$ is $hat{0}$. If $x$ is a complement of $a$, then we know that $x$ can't be $f$ or $e$, otherwise $xland a=a$. But we can't have $x$ be $b,c,$ or $d$ either, otherwise $xlor a=f$ if $x$ is $b$ or $c$ and $dlor a=a$. Thus, $a$ has no complements. The same argument applies to $b$ and $c$ by symmetry.
– Kevin Long
Nov 28 '18 at 18:33
add a comment |
I agree with you that $a,b,c$ have no complements. A complement of an element $x$ in a lattice is an element $y$ such that $xlor y=hat{0}$ and $xland y=hat{1}$. In this lattice, $e$ is $hat{1}$ and $d$ is $hat{0}$. If $x$ is a complement of $a$, then we know that $x$ can't be $f$ or $e$, otherwise $xland a=a$. But we can't have $x$ be $b,c,$ or $d$ either, otherwise $xlor a=f$ if $x$ is $b$ or $c$ and $dlor a=a$. Thus, $a$ has no complements. The same argument applies to $b$ and $c$ by symmetry.
– Kevin Long
Nov 28 '18 at 18:33
I agree with you that $a,b,c$ have no complements. A complement of an element $x$ in a lattice is an element $y$ such that $xlor y=hat{0}$ and $xland y=hat{1}$. In this lattice, $e$ is $hat{1}$ and $d$ is $hat{0}$. If $x$ is a complement of $a$, then we know that $x$ can't be $f$ or $e$, otherwise $xland a=a$. But we can't have $x$ be $b,c,$ or $d$ either, otherwise $xlor a=f$ if $x$ is $b$ or $c$ and $dlor a=a$. Thus, $a$ has no complements. The same argument applies to $b$ and $c$ by symmetry.
– Kevin Long
Nov 28 '18 at 18:33
I agree with you that $a,b,c$ have no complements. A complement of an element $x$ in a lattice is an element $y$ such that $xlor y=hat{0}$ and $xland y=hat{1}$. In this lattice, $e$ is $hat{1}$ and $d$ is $hat{0}$. If $x$ is a complement of $a$, then we know that $x$ can't be $f$ or $e$, otherwise $xland a=a$. But we can't have $x$ be $b,c,$ or $d$ either, otherwise $xlor a=f$ if $x$ is $b$ or $c$ and $dlor a=a$. Thus, $a$ has no complements. The same argument applies to $b$ and $c$ by symmetry.
– Kevin Long
Nov 28 '18 at 18:33
add a comment |
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I agree with you that $a,b,c$ have no complements. A complement of an element $x$ in a lattice is an element $y$ such that $xlor y=hat{0}$ and $xland y=hat{1}$. In this lattice, $e$ is $hat{1}$ and $d$ is $hat{0}$. If $x$ is a complement of $a$, then we know that $x$ can't be $f$ or $e$, otherwise $xland a=a$. But we can't have $x$ be $b,c,$ or $d$ either, otherwise $xlor a=f$ if $x$ is $b$ or $c$ and $dlor a=a$. Thus, $a$ has no complements. The same argument applies to $b$ and $c$ by symmetry.
– Kevin Long
Nov 28 '18 at 18:33